ABSTRACTS OF
TOPOLOGICAL GEOMETRODYNAMICS: AN OVERVIEW



PART I: A GENERAL OVERVIEW



Topological Geometrodynamics: What Might Be the First Principles?

A brief summary of various competing visions about the basic principles of quantum Topological Geometrodynamics (TGD) and about tensions between them is given with emphasis on the recent developments. These visions are following. Quantum physics as as classical spinor field geometry of the "world of classical worlds" consisting or light-like 3-surfaces of the 8-D imbedding space H=M4× CP2; zero energy ontology in which physical states correspond to physical events; TGD as almost topological quantum field theory for light-like 3-surfaces; physics as a generalized number theory with associativity defining the fundamental dynamical principle and involving a generalization of the number concept based on the fusion of real and p-adic number fields to a larger book like structure, the identification of real and various p-adic physics as algebraic completions of rational physics, and the notion of infinite prime; the identification of configuration space Clifford algebra elements as hyper-octonionic conformal fields with associativity condition implying what might be called number theoretic compacticitation; a generalization of quantum theory based on the introduction of hierarchy of Planck constants realized geometrically via a generalization of the notion of imbedding space H to a book like structure with pages which are coverings and orbifolds of H; the notion of finite measurement resolution realized in terms of inclusions of hyperfinite factors as the fundamental dynamical principle implying a generalization of S-matrix to M-matrix identified as Connes tensor product for positive and negative energy parts of zero energy states; two different kinds of extended super-conformal symmetries assignable to the light-cone of H and to the light-like 3-surfaces leading to a concrete construction recipe of M-matrix in terms of generalized Feynman diagrams having light-like 3-surfaces as lines and allowing to formulate generalized Einstein's equations in terms of coset construction.

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Overall View About the Evolution of TGD

This chapter provides a bird's eye view about TGD in its 27th birthday with the hope that this kind of summary might make it easier to follow the more technical representation provided by sub-sequent chapters. The geometrization of fundamental interactions assuming that space-times are representable as 4-surfaces of H=M4+× CP2 is wherefrom everything began. The two manners to understand TGD is TGD as a Poincare invariant theory of gravitation obtained by fusing special and general relativities, and TGD as a generalization of string model obtained my replacing 1-dimensional strings with 3-surfaces. The fusion of these approaches leads to the notion of the many-sheeted space-time.

The evolution of quantum TGD involves four threads which have become more and more entangled with each other. The first great vision was the reduction of the entire quantum physics (apart from quantum jump) to the geometry of classical spinor fields of the infinite-dimensional space of 3-surfaces in H, the great idea being that infinite-dimensional Kähler geometric existence and thus physics is unique from the requirement that it is free of infinities. The outcome is geometrization and generalization of the known structures of the quantum field theory and of string models.

The second thread is p-adic physics. p-Adic physics was initiated by more or less accidental observations about reduction of basic mass scale ratios to the ratios of square roots of Mersenne primes and leading to the p-adic thermodynamics explaining elementary particle mass scales and masses with an unexpected success. p-Adic physics turned eventually to be the physics of cognition and intentionality. Consciousness theory based ideas have led to a generalization of the notion of number obtained by gluing real numbers and various p-adic number fields along common rationals to a more general structure and implies that many-sheeted space-time contains also p-adic space-time sheets serving as space-time correlates of cognition and intentionality. The hypothesis that real and p-adic physics can be regarded as algebraic continuation of rational number based physics provides extremely strong constraints on the general structure of quantum TGD.

TGD inspired theory of consciousness can be seen as a generalization of quantum measurement theory replacing the notion of observer as an outsider with the notion of self. The detailed analysis of what happens in quantum jump have brought considerable understanding about the basic structure of quantum TGD itself. It seems that even quantum jump itself could be seen as a number theoretical necessity in the sense that state function reduction and state preparation by self measurements are necessary in order to reduce the generalized quantum state which is a formal superposition over components in different number fields to a state which contains only rational or finitely-extended rational entanglement identifiable as bound state entanglement. The number theoretical information measures generalizing Shannon entropy (always non-negative) are one of the important outcomes of consciousness theory combined with p-adic physics.

Physics as a generalized number theory is the fourth thread. The key idea is that the notion of divisibility could make sense also for literally infinite numbers and perhaps make them useful from the point of view of physicist. The great surprise was that the construction of infinite primes corresponds to the repeated quantization of a super-symmetric arithmetic quantum field theory. This led to the vision about physics as a generalized number theory involving infinite primes, integers, rationals and reals, as well as their quaternionic and octonionic counterparts. A further generalization is based on the generalization of the number concept already mentioned. Space-time surfaces could be regarded in this framework as concrete representations for infinite primes and integers, whereas the dimensions 8 and 4 for imbedding space and space-time surface could be seen as reflecting the dimensions of octonions and quaternions and their hyper counterparts obtained by multiplying imaginary units by √{-1. Also the dimension 2 emerges naturally as the maximal dimension of commutative sub-number field and relates to the ordinary conformal invariance central also for string models.

This chapter represents a overall view of classical TGD, a discussion of the p-adic concepts, a summary of the ideas generated by TGD inspired theory of consciousness, and the vision about physics as a generalized number theory. Also a proposal about how to predict the spectrum of Planck constants from quantum TGD is discussed.

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Overall View About Quantum TGD

This chapter provides a summary about quantum TGD. The discussions are based on the general vision that quantum states of the Universe correspond to the modes of classical spinor fields in the "world of the classical worlds" identified as the infinite-dimensional configuration space of 3-surfaces of H=M4×CP2 (more or less-equivalently, the corresponding 4-surfaces defining generalized Bohr orbits). The following topics are discussed on basis this vision.

1. Geometric ideas

TGD relies heavily on geometric ideas, which have gradually generalized during the years.

  1. The basic vision is that it is possible to reduce quantum theory to configuration space geometry and spinor structure. The geometrization of loop spaces inspires the idea that the mere existence of Riemann connection fixes configuration space Kähler geometry uniquely. Accordingly, configuration space can be regarded as a union of infinite-dimensional symmetric spaces labelled by zero modes labelling classical non-quantum fluctuating degrees of freedom. The huge symmetries of the configuration space geometry deriving from the light-likeness of 3-surfaces and from the special conformal properties of the boundary of 4-D light-cone would guarantee the maximal isometry group necessary for the symmetric space property. Quantum criticality is the fundamental hypothesis allowing to fix the Kähler function and thus dynamics of TGD uniquely. Quantum criticality leads to surprisingly strong predictions about the evolution of coupling constants.
  2. Configuration space spinors correspond to Fock states and anti-commutation relations for fermionic oscillator operators correspond to anti-commutation relations for the gamma matrices of the configuration space. Configuration space spinors define a von Neumann algebra known as hyper-finite factor of type II1 (HFFs). This has led to a profound understanding of quantum TGD. The outcome of this approach is that the exponents of Kähler function and Chern-Simons action are not fundamental objects but reduce to the Dirac determinant associated with the modified Dirac operator assigned to the light-like 3-surfaces.
  3. p-Adic mass calculations relying on p-adic length scale hypothesis led to an understanding of elementary particle masses using only super-conformal symmetries and p-adic thermodynamics. The need to fuse real physics and various p-adic physics to single coherent whole led to a generalization of the notion of number obtained by gluing together reals and p-adics together along common rationals and algebraics. The interpretation of p-adic space-time sheets is as correlates for cognition and intentionality. p-Adic and real space-time sheets intersect along common rationals and algebraics and the subset of these points defines what I call number theoretic braid in terms of which both configuration space geometry and S-matrix elements should be expressible. Thus one would obtain number theoretical discretization which involves no adhoc elements and is inherent to the physics of TGD.
  4. The work with HFFs combined with experimental input led to the notion of hierarchy of Planck constants interpreted in terms of dark matter. The hierarchy is realized via a generalization of the notion of imbedding space obtained by gluing infinite number of its variants along common lower-dimensional quantum critical sub-manifolds. This leads to the identification of number theoretical braids as points of partonic 2-surface which correspond to the minima of generalized eigenvalue of Dirac operator, a scalar field to which Higgs vacuum expectation is proportional to. Higgs vacuum expectation has thus a purely geometric interpretation. This leads to an explicit formula for the Dirac determinant. What is remarkable is that the construction gives also the 4-D space-time sheets associated with the light-like orbits of partonic 2-surfaces: they should correspond to preferred extremals of Kähler action. Thus hierarchy of Planck constants is now an essential part of construction of quantum TGD and of mathematical realization of the notion of quantum criticality.
  5. HFFs lead also to an idea about how entire TGD emerges from classical number fields, actually their complexifications. In particular, CP2 could be interpreted as a structure related to octonions. This would mean that TGD could be seen also as a generalized number theory.

2. Ideas related to the construction of S-matrix

The construction of S-matrix involves several ideas that have emerged during last years.

  1. Zero energy ontology motivated originally by TGD inspired cosmology means that physical states have vanishing net quantum numbers and are decomposable to positive and negative energy parts separated by a temporal distance characterizing the system as space-time sheet of finite size in time direction. The particle physics interpretation is as initial and final states of a particle reaction. S-matrix and density matrix are unified to the notion of M-matrix expressible as a product of square root of density matrix and of unitary S-matrix. Thermodynamics becomes therefore a part of quantum theory. One must distinguish M-matrix from U-matrix defined between zero energy states and analogous to S-matrix and characterizing the unitary process associated with quantum jump. U-matrix is most naturally related to the description of intentional action since in a well-defined sense it has elements between physical systems corresponding to different number fields.
  2. The notion of measurement resolution represented in terms of inclusions of HFFs is an essential element of the picture. Measurement resolution corresponds to the action of the included sub-algebra creating zero energy states in time scales shorter than the cutoff scale. This algebra effectively replaces complex numbers as coefficient fields and the condition that its action commutes with the M-matrix implies that M-matrix corresponds to Connes tensor product. Together with super-conformal symmetries this fixes possible M-matrices to a very high degree.
  3. Light-likeness of the basic fundamental objects implies that TGD is almost topological QFT so that the formulation in terms of category theoretical notions is expected to work. M-matrices form in a natural manner a functor from the category of cobordisms to the category of pairs of Hilbert spaces and this gives additional strong constraints on the theory.

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TGD and M-Theory

In this chapter a critical comparison of M-theory and TGD as two competing theories is carried out. Dualities and black hole physics are regarded as basic victories of M-theory.

a) The counterpart of electric magnetic duality plays an important role also in TGD and it has become clear that it might change the sign of Kähler coupling strength rather than leaving it invariant. The different signs would be related to different time orientations of the space-time sheets. This option is favored also by TGD inspired cosmology.

b) The AdS/CFT duality of Maldacena involved with the quantum gravitational holography has a direct counterpart in TGD with 3-dimensional causal determinants serving as holograms so that the construction of absolute minima of Kähler action reduces to a local problem.

c) The attempts to develop further the nebulous idea about space-time surfaces as quaternionic sub-manifolds of an octonionic imbedding space led to the realization of duality which could be called number theoretical spontaneous compactification. Space-time can be regarded equivalently as a hyper-quaternionic 4-surface in M8 with hyper-octonionic structure or as a 4-surface in M4× CP2. d) The duality of string models relating Kaluza-Klein quantum numbers with YM quantum numbers could generalize to a duality between 7-dimensional light like causal determinants of the imbedding space (analogs of "big bang") and 3-dimensional light like causal determinants of space-time surface (analogs of black hole horizons).

e) The notion of cotangent bundle of configuration space of 3-surfaces suggests the interpretation of the number-theoretical compactification as a wave-particle duality in infinite-dimensional context. Also the duality of hyper-quaternionic and co-hyper-quaternionic 4-surfaces could be understood analogously. These ideas generalize at the formal level also to the M-theory assuming that stringy configuration space is introduced. The existence of Kähler metric very probably does not allow dynamical target space.

In TGD framework black holes are possible but putting black holes and particles in the same basket seems to be mixing of apples with oranges. The role of black hole horizons is taken in TGD by 3-D light like causal determinants, which are much more general objects. Black hole-elementary particle correspondence and p-adic length scale hypothesis have already earlier led to a formula for the entropy associated with elementary particle horizon.

The recent findings from RHIC have led to the realization that TGD predicts black hole like objects in all length scales. They are identifiable as highly tangled magnetic flux tubes in Hagedorn temperature and containing conformally confined matter with a large Planck constant and behaving like dark matter in a macroscopic quantum phase. The fact that string like structures in macroscopic quantum states are ideal for topological quantum computation modifies dramatically the traditional view about black holes as information destroyers.

The discussion of the basic weaknesses of M-theory is motivated by the fact that the few predictions of the theory are wrong which has led to the introduction of anthropic principle to save the theory. The mouse as a tailor history of M-theory, the lack of a precise problem to which M-theory would be a solution, the hard nosed reductionism, and the censorship in Los Alamos archives preventing the interaction with competing theories could be seen as the basic reasons for the recent blind alley in M-theory.

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PART II:PHYSICS AS INFINITE-DIMENSIONAL SPINOR GEOMETRY IN THE WORLD OF CLASSICAL WORLDS



Classical TGD

In this chapter the classical field equations associated with the Kähler action are studied. The study of the extremals of the Kähler action has turned out to be extremely useful for the development of TGD. Towards the end of year 2003 quite dramatic progress occurred in the understanding of field equations and it seems that field equations might be in well-defined sense exactly solvable. Years later the understanding of quantum TGD at fundamental level deepened the understanding.

1. Preferred extremals and quantum criticality

The identification of preferred extremals of Kähler action defining counterparts of Bohr orbits has been one of the basic challenges of quantum TGD. By quantum classical correspondence the non-deterministic space-time dynamics should mimic the dissipative dynamics of the quantum jump sequence. It should also represent space-time correlate for quantum criticality.

The solution of the problem through the understanding of the implications number theoretical compactification and the realization of quantum TGD at fundamental level in terms of second quantization of induced spinor fields assigned to the modified Dirac action defined by Kähler action. Noether currents assignable to the modified Dirac equation are conserved only if the first variation of the modified Dirac operator DK defined by Kähler action vanishes. This is equivalent with the vanishing of the second variation of Kähler action -at least for the variations corresponding to dynamical symmetries having interpretation as dynamical degrees of freedom which are below measurement resolution and therefore effectively gauge symmetries.

The vanishing of the second variation in interior of X4(X3l) is what corresponds exactly to quantum criticality so that the basic vision about quantum dynamics of quantum TGD would lead directly to a precise identification of the preferred extremals. Something which I should have noticed for more than decade ago! The question whether these extremals correspond to absolute minima remains however open. The vanishing of second variations of preferred extremals suggests a generalization of catastrophe theory of Thom, where the rank of the matrix defined by the second derivatives of potential function defines a hierarchy of criticalities with the tip of bifurcation set of the catastrophe representing the complete vanishing of this matrix. In the recent case this theory would be generalized to infinite-dimensional context.

The space-time representation for dissipation comes from the interpretation of regions of space-time surface with Euclidian signature of induced metric as generalized Feynman diagrams (or equivalently the light-like 3-surfaces defining boundaries between Euclidian and Minkowskian regions). Dissipation would be represented in terms of Feynman graphs representing irreversible dynamics and expressed in the structure of zero energy state in which positive energy part corresponds to the initial state and negative energy part to the final state. Outside Euclidian regions classical dissipation should be absent and this indeed the case for the known extremals.

2. Hamilton-Jacobi structure

Most known extremals share very general properties. One of them is Hamilton-Jacobi structure meaning the possibility to assign to the extremal so called Hamilton-Jacobi coordinates. This means dual slicings of M4 by string world sheets and partonic 2-surfaces. Number theoretic compactification led years later to the same condition. This slicing allows a dimensional reduction of quantum TGD to Minkowksian and Euclidian variants of string model and allows to understand how Equivalence Principle is realized at space-time level. Also holography in the sense that the dynamics of 3-dimensional space-time surfaces reduces to that for 2-D partonic surfaces in a given measurement resolution follows. The construction of quantum TGD relies in essential manner to this property. CP2 type vacuum extremals do not possess Hamilton-Jaboci structure but this can be understood in the picture provided by number theoretical compactification.

3. Physical interpretation of extremals

The vanishing of Lorentz 4-force for the induced Kähler field means that the vacuum 4-currents are in a mechanical equilibrium and dissipation is absent except in the sense that the super-position of generalized Feynman graphs representing the zero energy state represents dissipation. Lorentz 4-force vanishes for all known solutions of field equations.

  1. The vanishing of the Lorentz 4-force in turn implies local covariant conservation of the ordinary energy momentum tensor. The corresponding condition is implied by Einstein's equations in General Relativity.

  2. The hypothesis would mean that the solutions of field equations are what might be called generalized Beltrami fields. The condition implies that vacuum currents can be non-vanishing only provided the dimension DCP2 of the CP2 projection of the space-time surface is less than four so that in the regions with DCP2=4, Maxwell's vacuum equations are satisfied.

  3. The hypothesis that Kähler current is proportional to a product of an arbitrary function φ of CP2 coordinates and of the instanton current generalizes Beltrami condition and reduces to it when electric field vanishes. Kähler current has vanishing divergence for DCP2 < 4, and Lorentz 4-force indeed vanishes. The remaining task would be the explicit construction of the imbeddings of these fields and the demonstration that field equations can be satisfied.

  4. Under additional conditions magnetic field reduces to what is known as Beltrami field. Beltrami fields are known to be extremely complex but highly organized structures. The natural conjecture is that topologically quantized many-sheeted magnetic and Z0 magnetic Beltrami fields and their generalizations serve as templates for the helical molecules populating living matter, and explain both chirality selection, the complex linking and knotting of DNA and protein molecules, and even the extremely complex and self-organized dynamics of biological systems at the molecular level.

  5. Beltrami fields appear in physical applications as asymptotic self organization patterns for which Lorentz force and dissipation vanish. Preferred extremal property abstracted to purely algebraic generalized Beltrami conditions would make sense also in the p-adic context as it should by number theoretic universality.

  6. As a consequence field equations can be reduced to algebraic conditions stating that energy momentum tensor and second fundamental form have no common components (this occurs also for minimal surfaces in string models) and only the conditions stating that Kähler current vanishes, is light-like, or proportional to instanton current, remain and define the remaining field equations. The conditions guaranteing topologization to instanton current can be solved explicitly. Solutions can be found also in the more general case when Kähler current is not proportional to instanton current. On basis of these findings there are strong reasons to believe that classical TGD is exactly solvable.

4. The dimension of CP2 projection as classifier for the fundamental phases of matter

The dimension DCP2 of CP2 projection of the space-time sheet encountered already in p-adic mass calculations classifies the fundamental phases of matter.

  1. For DCP2=4 empty space Maxwell equations would hold true. This phase is chaotic and analogous to de-magnetized phase. There is also a CP breaking associated with this phase. At least CP2 type vacuum extremals and their deformations represent this phase.

  2. DCP2=2 phase is analogous to ferromagnetic phase: highly ordered and relatively simple. In fact, this phase as such does not correspond to preferred extremals but only their small deformations obtained by topological condensation of CP2 type vacuum extremals representing elementary fermions at these extremals and by topological condensation of these extremals at larger space-time sheets creating wormhole contacts representing elementary bosons.

  3. DCP2=3 is the analog of spin glass and liquid crystal phases, extremely complex but highly organized by the properties of the generalized Beltrami fields. Also these extremals would represents ground states whose small deformations represent the phase. This phase is the boundary between chaos and order and corresponds to life emerging in the interaction of magnetic bodies with bio-matter. It is possible only in a finite temperature interval (note however the p-adic hierarchy of critical temperatures) and characterized by chirality just like life.

5. Specific extremals of Kähler action

The study of extremals of Kähler action represents more than decade old layer in the development of TGD.

  1. The huge vacuum degeneracy is the most characteristic feature of Kähler action (any 4-surface having CP2 projection which is Legendre sub-manifold is vacuum extremal, Legendre sub-manifolds of CP2 are in general 2-dimensional). This vacuum degeneracy is behind the spin glass analogy and leads to the p-adic TGD. As found in the second part of the book, various particle like vacuum extremals also play an important role in the understanding of the quantum TGD.

  2. The so called CP2 type vacuum extremals have finite, negative action and are therefore an excellent candidate for real particles whereas vacuum extremals with vanishing Kähler action are candidates for the virtual particles. These extremals have one dimensional M4 projection, which is light like curve but not necessarily geodesic and locally the metric of the extremal is that of CP2: the quantization of this motion leads to Virasoro algebra. Space-times with topology CP2#CP2#...CP2 are identified as the generalized Feynmann diagrams with lines thickened to 4-manifolds of "thickness" of the order of CP2 radius. The quantization of the random motion with light velocity associated with the CP2 type extremals in fact led to the discovery of Super Virasoro invariance, which through the construction of the configuration space geometry, becomes a basic symmetry of quantum TGD.

  3. There are also various non-vacuum extremals.

    1. String like objects, with string tension of same order of magnitude as possessed by the cosmic strings of GUTs, have a crucial role in TGD inspired model for the galaxy formation and in the TGD based cosmology.

    2. The so called massless extremals describe non-linear plane waves propagating with the velocity of light such that the polarization is fixed in given point of the space-time surface. Characteristic for TGD is the light like Kähler current: in the ordinary Maxwell theory vacuum gauge currents are not possible. This current serves as a source of coherent photons, which might play an important role in the quantum model of bio-system as a macroscopic quantum system.

    3. In the so called Maxwell's phase, ordinary Maxwell equations for the induced Kähler field are satisfied in an excellent approximation. A special case is provided by a radially symmetric extremal having an interpretation as the space-time exterior to a topologically condensed particle. The sign of the gravitational mass correlates with that of the Kähler charge and one can understand the generation of the matter antimatter asymmetry from the basic properties of this extremal. The possibility to understand the generation of the matter antimatter asymmetry directly from the basic equations of the theory gives strong support in favor of TGD in comparison to the ordinary EYM theories, where the generation of the matter antimatter asymmetry is still poorly understood.

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The Geometry of the World of the Classical Worlds

The topics of this chapter are the purely geometric aspects of the vision about physics as an infinite-dimensional Kähler geometry of the "world of classical worlds", with " classical world" identified either as 3-D surface of the unique Bohr orbit like 4-surface traversing through it. The non-determinism of Kähler action forces to generalize the notion of 3-surfaces so that unions of space-like surfaces with time like separations must be allowed. The considerations are restricted mostly to real context and the problems related to the p-adicization are discussed later.

There are two separate tasks involved.

  1. Provide configuration space of 3-surfaces with Kähler geometry which is consistent with 4-dimensional general coordinate invariance so that the metric is Diff4 degenerate. General coordinate invariance implies that the definition of metric must assign to a give 3-surface X3 a 4-surface as a kind of Bohr orbit X4(X3).

  2. Provide the configuration space with a spinor structure. The great idea is to identify configuration space gamma matrices in terms of super algebra generators expressible using second quantized fermionic oscillator operators for induced free spinor fields at the space-time surface assignable to a given 3-surface. The isometry generators and contractions of Killing vectors with gamma matrices would thus form a generalization of Super Kac-Moody algebra.

From the experience with loop spaces one can expect that there is no hope about existence of well-defined Riemann connection unless this space is union of infinite-dimensional symmetric spaces with constant curvature metric and simple considerations requires that Einstein equations are satisfied by each component in the union. The coordinates labeling these symmetric spaces are zero modes having interpretation as genuinely classical variables which do not quantum fluctuate since they do not contribute to the line element of the configuration space.

The construction of the Kähler structure involves also the identification of complex structure.

  1. Direct construction of Kähler function as action associated with a preferred Bohr orbit like extremal for some physically motivated action action leads to a unique result.

  2. Second approach is group theoretical and is based on a direct guess of isometries of the infinite-dimensional symmetric space formed by 3-surfaces with fixed values of zero modes. The group of isometries is generalization of Kac-Moody group obtained by replacing finite-dimensional Lie group with the group of symplectic transformations of δM4+×CP2, where δM4+ is the boundary of 4-dimensional future light-cone.

  3. Third approach is based on the conjecture that yhr vacuum functional of the theory identifiable as an exponent of Kähler function is expressible as a Dirac determinant. This approach leads to an explicit expression of configuration space metric in terms of finite number of eigenvalues assignable to the modified Dirac operator defined by Kähler action. The notion of number theoretical compactification and the known properties of extremals of Kähler action play key role in this approach.

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Configuration Space Spinor Structure

Quantum TGD should be reducible to the classical spinor geometry of the configuration space. In particular, physical states should correspond to the modes of the configuration space spinor fields. The immediate consequence is that configuration space spinor fields cannot, as one might naively expect, be carriers of a definite spin and unit fermion number. Concerning the construction of the configuration space spinor structure there are some important clues.

1. Geometrization of fermionic statistics in terms of configuration space spinor structure

The great vision has been that the second quantization of the induced spinor fields can be understood geometrically in terms of the configuration space spinor structure in the sense that the anti-commutation relations for configuration space gamma matrices require anti-commutation relations for the oscillator operators for free second quantized induced spinor fields.

  1. One must identify the counterparts of second quantized fermion fields as objects closely related to the configuration space spinor structure. Ramond model has as its basic field the anti-commuting field Gk(x), whose Fourier components are analogous to the gamma matrices of the configuration space and which behaves like a spin 3/2 fermionic field rather than a vector field. This suggests that the complexified gamma matrices of the configuration space are analogous to spin 3/2 fields and therefore expressible in terms of the fermionic oscillator operators so that their anti-commutativity naturally derives from the anti-commutativity of the fermionic oscillator operators.

    As a consequence, configuration space spinor fields can have arbitrary fermion number and there would be hopes of describing the whole physics in terms of configuration space spinor field. Clearly, fermionic oscillator operators would act in degrees of freedom analogous to the spin degrees of freedom of the ordinary spinor and bosonic oscillator operators would act in degrees of freedom analogous to the 'orbital' degrees of freedom of the ordinary spinor field.

  2. The classical theory for the bosonic fields is an essential part of the configuration space geometry. It would be very nice if the classical theory for the spinor fields would be contained in the definition of the configuration space spinor structure somehow. The properties of the modified massless Dirac operator associated with the induced spinor structure are indeed very physical. The modified massless Dirac equation for the induced spinors predicts a separate conservation of baryon and lepton numbers. The differences between quarks and leptons result from the different couplings to the CP2 Kähler potential. In fact, these properties are shared by the solutions of massless Dirac equation of the imbedding space.

  3. Since TGD should have a close relationship to the ordinary quantum field theories it would be highly desirable that the second quantized free induced spinor field would somehow appear in the definition of the configuration space geometry. This is indeed true if the complexified configuration space gamma matrices are linearly related to the oscillator operators associated with the second quantized induced spinor field on the space-time surface and/or its boundaries. There is actually no deep reason forbidding the gamma matrices of the configuration space to be spin half odd-integer objects whereas in the finite-dimensional case this is not possible in general. In fact, in the finite-dimensional case the equivalence of the spinorial and vectorial vielbeins forces the spinor and vector representations of the vielbein group SO(D) to have same dimension and this is possible for D=8-dimensional Euclidian space only. This coincidence might explain the success of 10-dimensional super string models for which the physical degrees of freedom effectively correspond to an 8-dimensional Euclidian space.

  4. It took a long time to realize that the ordinary definition of the gamma matrix algebra in terms of the anti-commutators {gA,gB} = 2gAB must in TGD context be replaced with {gAf,gB} = iJAB\per, where JAB denotes the matrix elements of the Kähler form of the configuration space. The presence of the Hermitian conjugation is necessary because configuration space gamma matrices carry fermion number. This definition is numerically equivalent with the standard one in the complex coordinates. The realization of this delicacy is necessary in order to understand how the square of the configuration space Dirac operator comes out correctly.

2. Modified Dirac equation for induced classical spinor fields

The construction of the spinor structure for the world of classical worlds (WCW) leads to the vision that second quantized modified Dirac equation codes for the entire quantum TGD. Among other things this would mean that Dirac determinant would define the vacuum functional of the theory having interpretation as the exponent of Kähler function of WCW and Kähler function would reduce to Kähler action for a preferred extremal of Kähler action.

Two alternative choices represented themselves as candidates for the modified Dirac action: either the 3-D Chern-Simons Dirac action or 4-D Kähler action. Eventually came the realization that the addition of a measurement interaction term to either Chern-Simons action or Kähler action is needed to resolve a bundle of conceptual problems. It took still some time to conclude that Kähler action with instanton term is the correct choice since the measurement interaction term assigned to Chern-Simons-Dirac action creates more problems that it solves. The resulting picture has rather profound implications for the understanding of quantum TGD.

3. Super-conformal symmetries

Modified Dirac action allows a rich structure of N=4 super-conformal symmetries. In particular, the generalized Kac-Moody symmetries leave corresponding X3-local isometries respecting the light-likeness condition. A rather detailed view about various aspects of super-conformal symmetries emerge leading to identification of fermionic anti-commutation relations and explicit expressions for configuration space gamma matrices and Kähler metric. This picture is consistent with the conditions posed by p-adic mass calculations.

Number theoretical considerations play a key role and lead to the picture in which effective discretization occurs so that partonic two-surface is effectively replaced by a discrete set of algebraic points belonging to the intersection of the real partonic 2-surface and its p-adic counterpart obeying the same algebraic equations. This implies effective discretization of super-conformal field theory giving N-point functions defining vertices via discrete versions of stringy formulas.

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PART III: ALGEBRAIC PHYSICS



TGD as a Generalized Number Theory I: p-Adicization Program

The vision about a number theoretic formulation of quantum TGD is based on the gradual accumulation of wisdom coming from different sources. The attempts to find a formulation allowing to understand real and p-adic physics as aspects of some more general scenario have been an important stimulus and generated a lot of, not necessarily mutually consistent ideas, some of which might serve as building blocks of the final formulation.

The first part of the 3-part chapter is devoted to the p-adicization program attempting to construct physics in various number fields as an algebraic continuation of physics in the field of rationals (or appropriate extension of rationals). The program involves in essential manner the generalization of number concept obtained by fusing reals and p-adic number fields to a larger structure by gluing them together along common rationals. Highly non-trivial number theoretic conjectures are an i outcome of the program.

1. Real and p-adic regions of the space-time as geometric correlates of matter and mind

The solutions of the equations determining space-time surfaces are restricted by the requirement that the imbedding space coordinates are real. When this is not the case, one might apply instead of a real completion with some rational-adic or p-adic completion: this is how rational-adic p-adic physics could emerge from the basic equations of the theory. One could interpret the resulting rational-adic or p-adic regions as geometrical correlates for 'mind stuff'.

p-Adic non-determinism implies extreme flexibility and therefore makes the identification of the p-adic regions as seats of cognitive representations very natural. Unlike real completion, p-adic completions preserve the information about the algebraic extension of rationals and algebraic coding of quantum numbers must be associated with 'mind like' regions of space-time. p-Adics and reals are in the same relationship as map and territory.

The implications are far-reaching and consistent with TGD inspired theory of consciousness: p-adic regions are present even at elementary particle level and provide some kind of model of 'self' and external world. In fact, p-adic physics must model the p-adic cognitive regions representing real elementary particle regions rather than elementary particles themselves!

2. The generalization of the notion of number

The unification of real physics of material work and p-adic physics of cognition and intentionality leads to the generalization of the notion of number field. Reals and various p-adic number fields are glued along their common rationals (and common algebraic numbers too) to form a fractal book like structure. Allowing all possible finite-dimensional extensions of p-adic numbers brings additional pages to this "Big Book".

At space-time level the book like structure corresponds to the decomposition of space-time surface to real and p-adic space-time sheets. This has deep implications for the view about cognition. For instance, two points infinitesimally near p-adically are infinitely distant in real sense so that cognition becomes a cosmic phenomenon.

3. p-Adicization by algebraic continuation

One general idea which results as an outcome of the generalized notion of number is the idea of a universal function continuable from a function mapping rationals to rationals or to a finite extension of rationals to a function in any number field. This algebraic continuation is analogous to the analytical continuation of a real analytic function to the complex plane. Rational functions with rational coefficients are obviously functions satisfying this constraint. Algebraic functions with rational coefficients satisfy this requirement if appropriate finite-dimensional algebraic extensions of p-adic numbers are allowed. Exponent function is such a function.

For instance, residue calculus might be generalized so that the value of an integral along the real axis could be calculated by continuing it instead of the complex plane to any number field via its values in the subset of rational numbers forming the rim of the book like structure having number fields as its pages. If the poles of the continued function in the finitely extended number field allow interpretation as real numbers it might be possible to generalize the residue formula. One can also imagine of extending residue calculus to any algebraic extension. An interesting situation arises when the poles correspond to extended p-adic rationals common to different pages of the "great book". Could this mean that the integral could be calculated at any page having the pole common. In particular, could a p-adic residue integral be calculated in the ordinary complex plane by utilizing the fact that in this case numerical approach makes sense.

Algebraic continuation is the basic tool of p-adicization program. Entire physics of the TGD Universe should be algebraically continuable to various number fields. Real number based physics would define the physics of matter and p-adic physics would describe correlates of cognition and intentionality. The basic stumbling block of this program is integration and algebraic continuation should allow to circumvent this difficulty. Needless to say, the requirement that the continuation exists must pose immensely tight constraints on the physics.

Due to the fact that real and p-adic topologies are fundamentally different, ultraviolet and infrared cutoffs in the set of rationals are unavoidable notions and correspond to a hierarchy of different physical phases on one hand and different levels of cognition on the other hand. Two types of cutoffs are predicted: p-adic length scale cutoff and a cutoff due to phase resolution. The latter cutoff seems to correspond naturally to the hierarchy of algebraic extensions of p-adic numbers and Beraha numbers Bn= 4cos2(π/n), n >=3 related closely to the hierarchy of quantum groups, braid groups, and II1 factors of von Neumann algebra. This cutoff hierarchy seems to relate closely to the hierarchy of cutoffs defined by the hierarchy of subalgebras of the super-canonical algebra defined by the hierarchy of sets (z1,...zn) , where zi are the first n non-trivial zeros of Riemann Zeta. Hence there are good hopes that the p-adicization program might unify apparently unrelated branches of mathematics.

4. Number theoretic democracy

The interpretation allows all finite-dimensional extensions of p-adic number fields and even infinite-P p-adics. The notion arithmetic quantum theory generalizes to include Gaussian and Eisenstein variants of infinite primes and corresponding arithmetic quantum field theories. Also the notion of p-adicity generalizes: it seems that one can indeed assign to Gaussian and Eisenstein primes what might be called G-adic and E-adic numbers. These number fields seem to be tailor made for modelling logarithmic spirals which represent the basic fractal like structures in a living matter and excitable media.

p-Adicization by algebraic continuation gives hopes of continuing quantum TGD from reals to various p-adic number fields. The existence of this continuation poses extremely strong constraints on theory and has already now inspired several number theoretic conjectures.

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TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts

This chapter is the second part of the multi-chapter devoted to the vision about TGD as a generalized number theory.

1. Hyper-quaterions and octonions

The original idea was that space-time surfaces could be regarded as four-surfaces in 8-D imbedding space with the property that the tangent spaces of these spaces can be locally regarded as 4- resp. 8-dimensional number fields of quaternions and octonions.

The difficulties caused by the Euclidian metric signature of the number theoretical norm have however forced to give up the original idea as such, and to introduce complexified octonions and quaternions resulting by extending quaternionic and octonionic algebra by adding imaginary units multiplied with √{-1. This spoils the number field property but the notion of prime is not lost. The sub-space of hyper-quaternions resp. -octonions is obtained from the algebra of ordinary quaternions and octonions by multiplying the imaginary part with √{-1. The transition is the number theoretical counterpart of the transition from Riemannian to pseudo-Riemannin geometry performed already in Special Relativity.

The problem is that H=M4× CP2 cannot be endowed with a hyper-octonionic manifold structure. Indeed, space-time surfaces are assumed to be hyper-quaternionic or co-hyper-quaternionic 4-surfaces of 8-dimensional Minkowski space M8 identifiable as the hyper-octonionic space HO. Since the hyper-quaternionic sub-spaces of HO with fixed complex structure are labelled by CP2, each (co)-hyper-quaternionic four-surface of HO defines a 4-surface of M4× CP2. One can say that the number-theoretic analog of spontaneous compactification occurs.

2. Space-time-surface as a hyper-quaternionic sub-manifold of hyper-octonionic imbedding space?

Space-time identified as a hyper-quaternionic (HQ) or co-hyper-quaternionic (coHQ) sub-manifold of the hyper-octonionic space in the sense that the tangent space or normal space of the space-time surface defines a hyper-quaternionic sub-algebra of the hyper-octonionic tangent space of H at each space-time point, looks an attractive idea. Second possibility is that the tangent space-algebra of the space-time surface is either associative or co-associative at each point.

The first guess would be that space-time surfaces are either hyper-quaternionic or co-hyper-quaternionic sub-manifolds of hyper-octonionic space HO=M8 with the property that complex structure is fixed and same at all points of space-time surface. This corresponds to a global selection of a preferred octonionic imaginary unit. The automorphisms leaving this selection invariant form group SU(3) identifiable as color group. The selections of hyper-quaternionic sub-space under this condition are parameterized by CP2. This means that each 4-surface in HO defines a 4-surface in M4× CP2 and one can speak about number-theoretic analog of spontaneous compactification having of course nothing to do with dynamics. It would be possible to make physics in two radically different geometric pictures: HO picture and H=M4× CP2 picture.

For a theoretical physicists of my generation it is easy to guess that the next step is to realize that it is possible to fix the preferred octonionic imaginary at each point of HO separately so that local S6=G2/SU(3) , or equivalently the local group G2 subject to SU(3) gauge invariance, characterizes the possible choices of hyper-quaternionic structure with a preferred imaginary unit. G2subset SO(7) is the automorphism group of octonions, and appears also in M-theory. This local choice has interpretation as a fixing of the plane of non-physical polarizations and rise to degeneracy which is a good candidate for the ground state degeneracy caused by the vacuum extremals.

HO--M4× CP2 duality allows to construct a foliation of HO by hyper-quaternionic space-time surfaces in terms of maps HO--> SU(3) satisfying certain integrability conditions guaranteing that the distribution of hyper-quaternionic planes integrates to a foliation by 4-surfaces. In fact, the freedom to fix the preferred imaginary unit locally extends the maps to HO--> G2 reducing to maps HO--> SU(3)×S6 in the local trivialization of G2. This foliation defines a four-parameter family of 4-surfaces in M4× CP2 for each local choice of the preferred imaginary unit. The dual of this foliation defines a 4-parameter family co-hyper-quaternionic space-time surfaces and it turns out that also these surfaces are needed.

Hyper-octonion analytic functions HO--> HO with real Taylor coefficients provide a physically motivated ansatz satisfying the integrability conditions. The basic reason is that hyper-octonion analyticity is not plagued by the complications due to non-commutativity and non-associativity. Indeed, this notion results also if the product is Abelianized by assuming that different octonionic imaginary units multiply to zero. A good candidate for the HO dynamics is free massless Dirac action with Weyl condition for an octonion valued spinor field using octonionic representation of γ matrices and coupled to the G2 gauge potential defined by the tensor 7×7 tensor product of the imaginary parts of spinor fields.

The basic conjecture is that HQ and coHQ surfaces correspond to preferred extremals of Kähler action. This conjecture has several variants. It could be that only asymptotic behavior corresponds to HQ analytic function but that HQ and coHQ is a generic property. It could also be that maxima of Kähler function correspond to this kind of 4-surfaces. The encouraging hint is the fact that Hamilton-Jacobi coordinates appear naturally also in the construction of general solutions of field equations.

3. The notion of Kähler calibration

Calibration is a closed p-form, whose value for a given p-plane is not larger than its volume in the induced metric. What is important that if it is maximum for tangent planes of p-sub-manifold, minimal surface with smallest volume in its homology equivalence class results.

The idea of Kähler calibration is based on a simple observation. The octonionic spinor field defines a map M8--> H=M4× CP2 allowing to induce metric and Kähler form of H to M8. Also Kähler action is well defined for the local hyper-quaternion plane.

The idea is that the non-closed 4-form associated the wedge product of unit tangent vectors of hyper-quaternionic plane in M8 and saturating to volume for it becomes closed by multiplication with Kähler action density LK. If LK is minimal for hyper-quaternion plane, hyper-quaternionic manifolds define extremals of Kähler action for which the magnitudes of positive and negative contributions to the action are separately minimized.

This variational principle is not equivalent with absolute minimization of Kähler action. Rather, Universe would do its best to save energy, being as near as possible to vacuum. Also vacuum extremals would become physically relevant (they carry non-vanishing density gravitational energy). The non-determinism of the vacuum extremals would have an interpretation in terms of the ability of Universe to engineer itself.

The attractiveness of the number theoretical variational principle from the point of calculability of TGD would be that the initial values for the time derivatives of the imbedding space coordinates at X3 at light-like 7-D causal determinant could be computed by requiring that the energy of the solution is minimized. This could mean a computerizable construction of Kähler function.

In coHQ phase Universe would obviously maximize fluctuations and contrasts in accordance with quantum criticality. One might say that these two phases give Universe kind of hawk-dove polarity.

One can assign to a given 3-surface both HQ and cHQ 4-surface in the generic case and the equivalence of descriptions requires that corresponding Kähler functions differ by the real part of a holomorphic function of CH coordinates.

4. Generalizing the notion of HO-H duality to quantum level

The obvious question is how the HO-H duality could generalize to quantum level. Number theoretical considerations combined with the general vision about generalized Feynman diagrams as a generalization of braid diagrams lead to general formulas for vertices in HO picture.

Simple arguments lead to the conclusion that strict duality can make sense only if the hyper-octonionic spinor field is second quantized in some sense. One can imagine two, not necessarily mutually exclusive, manners to quantize.

a) The construction of the spinor structure for the configuration space of 3-surfaces in HO forces to conclude that HO spinor fields induced to X4subset HO are second quantized as usual and define configuration space γ matrices as super generators. The classical real-analytic HO spinor fields would represent analogs of zero modes of H spinor fields. The second quantized part of hyper-octonionic spinor fields induced to X4subset HO would have 1+1+3+\overline{3 decomposition having interpretation in terms of quarks and leptons and second quantized oscillator operators would commute with hyper-octonionic units. The detailed realization of HO-H duality suggests that the induced spinor fields at X4subset H resp. X4subset HO are restrictions of H resp. HO spinor fields. This would hold for zero modes and could hold for second quantized part too.

b) The original idea was that the real Laurent coefficients correspond to a complete set of mutually commuting Hermitian operators having interpretation as observables. This is not enough for configuration space geometry but is favored by quantum classical correspondence. Space-time concept would be well defined only for the eigen states of these operators and physical states are mapped to space-time surfaces. The Hermitian operators would naturally correspond to the state space spanned by super Kac-Moody and super-canonical algebras, and quantum states would have precise space-time counterparts in accordance with quantum-classical correspondence.

The regions inside which the power series representing HO analytic function and matrix elements of G2 rotation converge are identified as counterparts of maximal deterministic regions of the space-time surface. The Hermitian operators replacing Laurent coefficients are assumed to commute inside these regions identifiable also as coherence regions for the generalized Schrödinger amplitude represented by the HO spinor field.

By quantum classical correspondence these regions would be correlates for the final states of quantum jumps. The space-like 3-D causal determinants X3 bounding adjacent regions of this kind represent quantum jumps. The hyper-octonionic part of the inner of the hyper-octonionic spinor fields at the two sides of the discontinuity defined as an integral over X3 would give a number identifiable as complex number when imaginary unit is identified appropriately. The inner product would be identified as a representation of S-matrix element for an internal transition of particle represented by 3-surface, that is 2-vertex.

For the generalized Feynman diagrams n -vertex corresponds to a fusion of n 4-surfaces along their ends at X3. 3-vertex can be represented number theoretically as a triality of three hyper-octonion spinors integrated over the 3-surface in question. Higher vertices can be defined as composite functions of triality with a map (h1,h2)--> \overline{h3 defined by octonionic triality and by duality given by the inner product. More concretely, m+n vertex corresponds in HO picture to the inner product for the local hyper-octonionic products of m outgoing and n incoming hyper-octonionic spinor fields integrated over the 3-surface defining the vertex. Both 2-vertices representing internal transitions and n>2 vertices are completely fixed. This should give some idea about the power of the number theoretical vision.

One can raise objections against the need for non-conventional quantization. The number theoretic prescription does not apply to the second quantized parts of HO spinor fields and S-matrix elements can be constructed using them so that two equivalent prescriptions of S-matrix would emerge. On the other hand, TGD inspired quantum measurement theory suggests dual codings S-matrix elements based on either quantum states or classical observables (zero modes) in 1-1 correspondence with them.

5. Does TGD reduce to 8-D WZW string model?

Conservation laws suggests that in the case of non-vacuum extremals the dynamics of the local G2 automorphism is dictated by field equations of some kind. The experience with WZW model suggests that in the case of non-vacuum extremals G2 element could be written as a product g=gL(h)g-1R(h*) of hyper-octonion analytic and anti-analytic complexified G2 elements. g would be determined by the data at hyper-complex 2-surface for which the tangent space at a given point is spanned by real unit and preferred hyper-octonionic unit. Also Dirac action would be naturally restricted to this surface. The theory would reduce in HO picture to 8-D WZW string model both classically and quantally since vertices would reduce to integrals over 1-D curves.

The interpretation of generalized Feynman diagrams in terms of generalized braid/ribbon diagrams and the unique properties of G2 provide further support for this picture. In particular, G2 is the lowest-dimensional Lie group allowing to realize full-powered topological quantum computation based on generalized braid diagrams and using the lowest level k=1 Kac Moody representation. Even if this reduction would occur only in special cases, such as asymptotic solutions for which Lorentz Kähler force vanishes or maxima of Kähler function, it would mean enormous simplification of the theory.

6. Why (co-)hyper-quaternionicity corresponds to the minimization of Kähler action?

The resulting over all picture leads also to a considerable understanding concerning the basic questions why (co)-hyper-quaternionic 4-surfaces define extrema of Kähler action and why WZW strings would provide a dual for the description using Kähler action. The answer boils down to the realization that the extrema of Kähler action minimize complexity, also algebraic complexity, in particular non-commutativity. A measure for non-commutativity with a fixed preferred hyper-octonionic imaginary unit is provided by the commutator of 3 and 3bar parts of the hyper-octonion spinor field defining an antisymmetric tensor in color octet representation: very much like color gauge field. Color action is a natural measure for the non-commutativity minimized when the tangent space algebra closes to complexified quaternionic, instead of complexified octonionic, algebra. On the other hand, Kähler action is nothing but color action for classical color gauge field defined by projections of color Killing vector fields. That WZW + Dirac action for hyper-octonionic strings would correspond to Kähler action would in turn be the TGD counterpart for the proposed string-YM dualities.

7. Various dualities and their physical counterparts

HO-H duality is only one representative in a family of dualities characterizing TGD. It is not equivalent with HQ-coHQ duality, which seems however to be equivalent with the electric-magnetic duality known for long. This duality relates descriptions based on space-like partonic 2-surfaces and time-like string orbits. HO-H and HQ-coHQ dualities seem to be closely correlated in the sense that HO picture is natural in HQ phase and H picture in coHQ phase.

At configuration space level HO-H duality means roughly following. In H picture spin and ew spin are spin-like quantum numbers whereas color is orbital quantum number and cannot be seen at space-time level directly. In HO picture the roles of these quantum numbers are changed. One could say that HO-H duality acts as a super-symmetry permuting spin and orbital degrees of freedom of configuration space spinor fields. This duality allows a surprisingly detailed understanding of almost paradoxical dualities of hadron physics, and also explains proton spin crisis from first principles.

It seems possible to interpret HO-H and HQ-coHQ dualities as analogs of wave-particle duality in the infinite-dimensional context. For HO-H duality the cotangent bundle of configuration space CH would be the unifying notion. Position q in CH would be represented by 3-surface whereas canonical momentum p would correspond to the same 3-surface but as a surface in CHO with induced metric and Kähler structure inherited from HO defining the tangent space of H. The notion of stringy configuration space might allow to understand also M-theory dualities in this manner.

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TGD as a Generalized Number Theory III: Infinite Primes

Infinite primes are besides p-adicization and the representation of space-time surface as a hyper-quaternionic sub-manifold of hyper-octonionic space, basic pillars of the vision about TGD as a generalized number theory and will be discussed in the third part of the multi-chapter devoted to the attempt to articulate this vision as clearly as possible.

1. Why infinite primes are unavoidable

Suppose that 3-surfaces could be characterized by p-adic primes characterizing their effective p-adic topology. p-Adic unitarity implies that each quantum jump involves unitarity evolution U followed by a quantum jump. Simple arguments show that the p-adic prime characterizing the 3-surface representing the entire universe increases in a statistical sense. This leads to a peculiar paradox: if the number of quantum jumps already occurred is infinite, this prime is most naturally infinite. On the other hand, if one assumes that only finite number of quantum jumps have occurred, one encounters the problem of understanding why the initial quantum history was what it was. Furthermore, since the size of the 3-surface representing the entire Universe is infinite, p-adic length scale hypothesis suggest also that the p-adic prime associated with the entire universe is infinite.

These arguments motivate the attempt to construct a theory of infinite primes and to extend quantum TGD so that also infinite primes are possible. Rather surprisingly, one can construct what might be called generating infinite primes by repeating a procedure analogous to a quantization of a super symmetric quantum field theory. At given level of hierarchy one can identify the decomposition of space-time surface to p-adic regions with the corresponding decomposition of the infinite prime to primes at a lower level of infinity: at the basic level are finite primes for which one cannot find any formula.

2. Two views about the role of infinite primes and physics in TGD Universe

Two different views about how infinite primes, integers, and rationals might be relevant in TGD Universe have emerged.

a) The first view is based on the idea that infinite primes characterize quantum states of the entire Universe. 8-D hyper-octonions make this correspondence very concrete since 8-D hyper-octonions have interpretation as 8-momenta. By quantum-classical correspondence also the decomposition of space-time surfaces to p-adic space-time sheets should be coded by infinite hyper-octonionic primes. Infinite primes could even have a representation as hyper-quaternionic 4-surfaces of 8-D hyper-octonionic imbedding space.

b) The second view is based on the idea that infinitely structured space-time points define space-time correlates of mathematical cognition. The mathematical analog of Brahman=Atman identity would however suggest that both views deserve to be taken seriously.

3. Infinite primes and infinite hierarchy of second quantizations

The discovery of infinite primes suggested strongly the possibility to reduce physics to number theory. The construction of infinite primes can be regarded as a repeated second quantization of a super-symmetric arithmetic quantum field theory. Later it became clear that the process generalizes so that it applies in the case of quaternionic and octonionic primes and their hyper counterparts. This hierarchy of second quantizations means enormous generalization of physics to what might be regarded a physical counterpart for a hierarchy of abstractions about abstractions about.. The ordinary second quantized quantum physics corresponds only to the lowest level infinite primes. This hierarchy can be identified with the corresponding hierarchy of space-time sheets of the many-sheeted space-time.

One can even try to understand the quantum numbers of physical particles in terms of infinite primes. In particular, the hyper-quaternionic primes correspond four-momenta and mass squared is prime valued for them. The properties of 8-D hyper-octonionic primes motivate the attempt to identify the quantum numbers associated with CP2 degrees of freedom in terms of these primes. The representations of color group SU(3) are indeed labelled by two integers and the states inside given representation by color hyper-charge and iso-spin.

4. Infinite primes as a bridge between quantum and classical

An important stimulus came from the observation stimulated by algebraic number theory. Infinite primes can be mapped to polynomial primes and this observation allows to identify completely generally the spectrum of infinite primes whereas hitherto it was possible to construct explicitly only what might be called generating infinite primes.

This in turn led to the idea that it might be possible represent infinite primes (integers) geometrically as surfaces defined by the polynomials associated with infinite primes (integers).

Obviously, infinite primes would serve as a bridge between Fock-space descriptions and geometric descriptions of physics: quantum and classical. Geometric objects could be seen as concrete representations of infinite numbers providing amplification of infinitesimals to macroscopic deformations of space-time surface. We see the infinitesimals as concrete geometric shapes!

5. Various equivalent characterizations of space-times as surfaces

One can imagine several number-theoretic characterizations of the space-time surface.

  1. The approach based on octonions and quaternions suggests that space-time surfaces correspond to associative, or equivalently, hyper-quaternionic surfaces of hyper-octonionic imbedding space HO. Also co-associative, or equivalently, co-hyper-quaternionic surfaces are possible. These foliations can be mapped in a natural manner to the foliations of H=M^4\times CP_2 by space-time surfaces which are identified as preferred extremals of the K\"ahler action (absolute minima or maxima for regions of space-time surface in which action density has definite sign). These views are consistent if hyper-quaternionic space-time surfaces correspond to so called K\"ahler calibrations \cite{visionb}.

  2. Hyper-octonion real-analytic surfaces define foliations of the imbedding space to hyper-quaternionic 4-surfaces and their duals to co-hyper-quaternionic 4-surfaces representing space-time surfaces.

  3. Rational infinite primes can be mapped to rational functions of n arguments. For hyper-octonionic arguments non-associativity makes these functions poorly defined unless one assumes that arguments are related by hyper-octonion real-analytic maps so that only single independent variable remains. These hyper-octonion real-analytic functions define foliations of HO to space-time surfaces if b) holds true.

The challenge of optimist is to prove that these characterizations are equivalent.

6. The representation of infinite primes as 4-surfaces

The difficulties caused by the Euclidian metric signature of the number theoretical norm forced to give up the idea that space-time surfaces could be regarded as quaternionic sub-manifolds of octonionic space, and to introduce complexified octonions and quaternions resulting by extending quaternionic and octonionic algebra by adding imaginary units multiplied with √{-1. This spoils the number field property but the notion of prime is not lost. The sub-space of hyper-quaternions resp.-octonions is obtained from the algebra of ordinary quaternions and octonions by multiplying the imaginary part with √-1. The transition is the number theoretical counterpart for the transition from Riemannian to pseudo-Riemannian geometry performed already in Special Relativity.

The commutative √-1 relates naturally to the algebraic extension of rationals generalized to an algebraic extension of rational quaternions and octonions and conforms with the vision about how quantum TGD could emerge from infinite dimensional Clifford algebra identifiable as a hyper-finite factor of type II1.

The notions of hyper-quaternionic and octonionic manifold make sense but it is implausible that H=M4× CP2 could be endowed with a hyper-octonionic manifold structure. Indeed, space-time surfaces are assumed to be hyper-quaternionic or co-hyper-quaternionic 4-surfaces of 8-dimensional Minkowski space M8 identifiable as the hyper-octonionic space HO. Since the hyper-quaternionic sub-spaces of HO with a fixed complex structure are labelled by CP2, each (co)-hyper-quaternionic four-surface of HO defines a 4-surface of M4× CP2. One can say that the number-theoretic analog of spontaneous compactification occurs.

Any hyper-octonion analytic function HO--> HO defines a function g: HO--> SU(3) acting as the group of octonion automorphisms leaving a selected imaginary unit invariant, and g in turn defines a foliation of HO and H=M4× CP2 by space-time surfaces. The selection can be local which means that G2 appears as a local gauge group.

Since the notion of prime makes sense for the complexified octonions, it makes sense also for the hyper-octonions. It is possible to assign to infinite prime of this kind a hyper-octonion analytic polynomial P: HO--> HO and hence also a foliation of HO and H=M4× CP2 by 4-surfaces. Therefore space-time surface could be seen as a geometric counterpart of a Fock state. The assignment is not unique but determined only up to an element of the local octonionic automorphism group G2 acting in HO and fixing the local choices of the preferred imaginary unit of the hyper-octonionic tangent plane. In fact, a map HO--> S6 characterizes the choice since SO(6) acts effectively as a local gauge group.

The construction generalizes to all levels of the hierarchy of infinite primes if one poses the associativity requirement implying that hyper-octonionic variables are related by hyper-octonion real-analytic maps, and produces also representations for integers and rationals associated with hyper-octonionic numbers as space-time surfaces. By the effective 2-dimensionality naturally associated with infinite primes represented by real polynomials 4-surfaces are determined by data given at partonic 2-surfaces defined by the intersections of 3-D and 7-D light-like causal determinants. In particular, the notions of genus and degree serve as classifiers of the algebraic geometry of the 4-surfaces. The great dream is of course to prove that this construction yields the solutions to the absolute minimization of Kähler action.

7. Generalization of ordinary number fields: infinite primes and cognition

Both fermions and p-adic space-time sheets are identified as correlates of cognition in TGD Universe. The attempt to relate these two identifications leads to a rather concrete model for how bosonic generators of super-algebras correspond to either real or p-adic space-time sheets (actions and intentions) and fermionic generators to pairs of real space-time sheets and their p-adic variants obtained by algebraic continuation (note the analogy with fermion hole pairs).

The introduction of infinite primes, integers, and rationals leads also to a generalization of real numbers since an infinite algebra of real units defined by finite ratios of infinite rationals multiplied by ordinary rationals which are their inverses becomes possible. These units are not units in the p-adic sense and have a finite p-adic norm which can be differ from one. This construction generalizes also to the case of hyper- quaternions and -octonions although non-commutativity and in case of octonions also non-associativity pose technical problems to which the reduction to ordinary rational is simplest cure which would however allow interpretation as decomposition of infinite prime to hyper-octonionic lower level constituents. Obviously this approach differs from the standard introduction of infinitesimals in the sense that sum is replaced by multiplication meaning that the set of real units becomes infinitely degenerate.

Infinite primes form an infinite hierarchy so that the points of space-time and imbedding space can be seen as infinitely structured and able to represent all imaginable algebraic structures. Certainly counter-intuitively, single space-time point is even capable of representing the quantum state of the entire physical Universe in its structure. For instance, in the real sense surfaces in the space of units correspond to the same real number 1, and single point, which is structure-less in the real sense could represent arbitrarily high-dimensional spaces as unions of real units.

One might argue that for the real physics this structure is completely invisible and is relevant only for the physics of cognition. On the other hand, one can consider the possibility of mapping the configuration space and configuration space spinor fields to the number theoretical anatomies of a single point of imbedding space so that the structure of this point would code for the world of classical worlds and for the quantum states of the Universe. Quantum jumps would induce changes of configuration space spinor fields interpreted as wave functions in the set of number theoretical anatomies of single point of imbedding space in the ordinary sense of the word, and evolution would reduce to the evolution of the structure of a typical space-time point in the system. Physics would reduce to space-time level but in a generalized sense. Universe would be an algebraic hologram, and there is an obvious connection both with Brahman=Atman identity of Eastern philosophies and Leibniz's notion of monad.

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PART IV: HYPERFINITE FACTORS OF TYPE II1 AND HIERARCHY OF PLANCK CONSTANTS



Was von Neumann Right After All?

The work with TGD inspired model for quantum computation led to the realization that von Neumann algebras, in particular hyper-finite factors, could provide the mathematics needed to develop a more explicit view about the construction of M-matrix generalizing the notion of S-matrix in zero energy ontology. In this chapter I will discuss various aspects of hyper-finite factors and their possible physical interpretation in TGD framework. The original discussion has transformed during years from free speculation reflecting in many aspects my ignorance about the mathematics involved to a more realistic view about the role of these algebras in quantum TGD.

Hyper-finite factors in quantum TGD

The following argument suggests that von Neumann algebras known as hyper-finite factors (HFFs) of type III1 appearing in relativistic quantum field theories provide also the proper mathematical framework for quantum TGD.

  1. The Clifford algebra of the infinite-dimensional Hilbert space is a von Neumann algebra known as HFF of type II1. There also the Clifford algebra at a given point (light-like 3-surface) of world of classical worlds (WCW) is therefore HFF of type II1. If the fermionic Fock algebra defined by the fermionic oscillator operators assignable to the induced spinor fields (this is actually not obvious!) is infinite-dimensional it defines a representation for HFF of type II1. Super-conformal symmetry suggests that the extension of the Clifford algebra defining the fermionic part of a super-conformal algebra by adding bosonic super-generators representing symmetries of WCW respects the HFF property. It could however occur that HFF of type II results.

  2. WCW is a union of sub-WCWs associated with causal diamonds (CD) defined as intersections of future and past directed light-cones. One can allow also unions of CDs and the proposal is that CDs within CDs are possible. Whether CDs can intersect is not clear.

  3. The assumption that the M4 proper distance a between the tips of CD is quantized in powers of 2 reproduces p-adic length scale hypothesis but one must also consider the possibility that a can have all possible values. Since SO(3) is the isotropy group of CD, the CDs associated with a given value of a and with fixed lower tip are parameterized by the Lobatchevski space L(a)=SO(3,1)/SO(3). Therefore the CDs with a free position of lower tip are parameterized by M4×L(a). A possible interpretation is in terms of quantum cosmology with a identified as cosmic time . Since Lorentz boosts define a non-compact group, the generalization of so called crossed product construction strongly suggests that the local Clifford algebra of WCW is HFF of type III1. If one allows all values of a, one ends up with M4×M4+ as the space of moduli for WCW.

  4. An interesting special aspect of 8-dimensional Clifford algebra with Minkowski signature is that it allows an octonionic representation of gamma matrices obtained as tensor products of unit matrix 1 and 7-D gamma matrices γk and Pauli sigma matrices by replacing 1 and γk by octonions. This inspires the idea that it might be possible to end up with quantum TGD from purely number theoretical arguments. This seems to be the case. One can start from a local octonionic Clifford algebra in M8. Associativity condition is satisfied if one restricts the octonionic algebra to a subalgebra associated with any hyper-quaternionic and thus 4-D sub-manifold of M8. This means that the modified gamma matrices associated with the Kähler action span a complex quaternionic sub-space at each point of the sub-manifold. This associative sub-algebra can be mapped a matrix algebra. Together with M8-H duality this leads automatically to quantum TGD and therefore also to the notion of WCW and its Clifford algebra which is however only mappable to an associative algebra and thus to HFF of type II1.

2. Hyper-finite factors and M-matrix

HFFs of type III1 provide a general vision about M-matrix.

  1. The factors of type III allow unique modular automorphism Dit (fixed apart from unitary inner automorphism). This raises the question whether the modular automorphism could be used to define the M-matrix of quantum TGD. This is not the case as is obvious already from the fact that unitary time evolution is not a sensible concept in zero energy ontology.

  2. Concerning the identification of M-matrix the notion of state as it is used in theory of factors is a more appropriate starting point than the notion modular automorphism but as a generalization of thermodynamical state is certainly not enough for the purposes of quantum TGD and quantum field theories (algebraic quantum field theorists might disagree!). Zero energy ontology requires that the notion of thermodynamical state should be replaced with its "complex square root" abstracting the idea about M-matrix as a product of positive square root of a diagonal density matrix and a unitary S-matrix. This generalization of thermodynamical state -if it exists- would provide a firm mathematical basis for the notion of M-matrix and for the fuzzy notion of path integral.

  3. The existence of the modular automorphisms relies on Tomita-Takesaki theorem, which assumes that the Hilbert space in which HFF acts allows cyclic and separable vector serving as ground state for both HFF and its commutant. The translation to the language of physicists states that the vacuum is a tensor product of two vacua annihilated by annihilation oscillator type algebra elements of HFF and creation operator type algebra elements of its commutant isomorphic to it. Note however that these algebras commute so that the two algebras are not hermitian conjugates of each other. This kind of situation is exactly what emerges in zero energy ontology: the two vacua can be assigned with the positive and negative energy parts of the zero energy states entangled by M-matrix.

  4. There exists infinite number of thermodynamical states related by modular automorphisms. This must be true also for their possibly existing "complex square roots". Physically they would correspond to different measurement interactions giving rise to Kähler functions of WCW differing only by a real part of holomorphic function of complex coordinates of WCW and arbitrary function of zero mode coordinates and giving rise to the same Kähler metric of WCW.

The concrete construction of M-matrix utilizing the idea of bosonic emergence (bosons as fermion anti-fermion pairs at opposite throats of wormhole contact) meaning that bosonic propagators reduce to fermionic loops identifiable as wormhole contacts leads to generalized Feynman rules for M-matrix in which modified Dirac action containing measurement interaction term defines stringy propagators. This M-matrix should be consistent with the above proposal.

3. Connes tensor product as a realization of finite measurement resolution

The inclusions N subset M of factors allow an attractive mathematical description of finite measurement resolution in terms of Connes tensor product but do not fix M-matrix as was the original optimistic belief.

  1. In zero energy ontology N would create states experimentally indistinguishable from the original one. Therefore N takes the role of complex numbers in non-commutative quantum theory. The space M/N would correspond to the operators creating physical states modulo measurement resolution and has typically fractal dimension given as the index of the inclusion. The corresponding spinor spaces have an identification as quantum spaces with non-commutative N-valued coordinates.

  2. This leads to an elegant description of finite measurement resolution. Suppose that a universal M-matrix describing the situation for an ideal measurement resolution exists as the idea about square root of state encourages to think. Finite measurement resolution forces to replace the probabilities defined by the M-matrix with their N-"averaged" counterparts. The "averaging" would be in terms of the complex square root of N-state and a direct analog of functionally or path integral over the degrees of freedom below measurement resolution defined by (say) length scale cutoff.

  3. One can construct also directly M-matrices satisfying the measurement resolution constraint. The condition that N acts like complex numbers on M-matrix elements as far as N-"averaged" probabilities are considered is satisfied if M-matrix is a tensor product of M-matrix in M/N interpreted as finite-dimensional space with a projection operator to N. The condition that N averaging in terms of a complex square root of N state produces this kind of M-matrix poses a very strong constraint on M-matrix if it is assumed to be universal (apart from variants corresponding to different measurement interactions).

4. Quantum spinors and fuzzy quantum mechanics

The notion of quantum spinor leads to a quantum mechanical description of fuzzy probabilities. For quantum spinors state function reduction cannot be performed unless quantum deformation parameter equals to q=1. The reason is that the components of quantum spinor do not commute: it is however possible to measure the commuting operators representing moduli squared of the components giving the probabilities associated with 'true' and 'false'. The universal eigenvalue spectrum for probabilities does not in general contain (1,0) so that quantum qbits are inherently fuzzy. State function reduction would occur only after a transition to q=1 phase and decoherence is not a problem as long as it does not induce this transition.

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Does TGD Predict the Spectrum of Planck Constants?

The quantization of Planck constant has been the basic them of TGD since 2005. The basic idea was stimulated by the finding of Nottale that planetary orbits could be seen as Bohr orbits with enormous value of Planck constant given by hbargr = GM1M2/v0, v0 @ 2-11 for the inner planets. This inspired the ideas that quantization is due to a condensation of ordinary matter around dark matter concentrated near Bohr orbits and that dark matter is in macroscopic quantum phase in astrophysical scales. The second crucial empirical input were the anomalies associated with living matter. The revised version of the chapter represents the vision about quantization of Planck constants from a perspective given by almost five years work with the idea. A very concise summary about the situation is as follows.

  1. The hierarchy of Planck constants cannot be realized without generalizing the notions of imbedding space M4×CP2 and space-time since particles with different values of Planck constant cannot appear in the same interaction vertex. This suggests some kind of book like structure for both M4 and CP2 factors of the generalized imbedding space is suggestive.

  2. Schrödinger equation suggests that Planck constant corresponds to a scaling factor of M4 metric whose value labels different pages of the book. The scaling of M4 coordinate so that original metric results in M4 factor is possible so that the scaling of hbar corresponds to the scaling of the size of causal diamond CD defined as the intersection of future and past directed light-cones. The light-like 3-surfaces having their 2-D and light-boundaries of CD are in a key role in the realization of zero energy states. The infinite-D spaces formed by these 3-surfaces define the fundamental sectors of the configuration space (world of classical worlds). Since the scaling of CD does not simply scale space-time surfaces, the coding of radiative corrections to the geometry of space-time sheets becomes possible and Kähler action can be seen as expansion in powers of hbar/hbar0.

  3. Quantum criticality of TGD Universe is one of the key postulates of quantum TGD. The most important implication is that Kähler coupling strength is analogous to critical temperature. The exact realization of quantum criticality would be in terms of critical sub-manifolds of M4 and CP2 common to all sectors of the generalized imbedding space. Quantum criticality would mean that the two kinds of number theoretic braids assignable to M4 and CP2 projections of the partonic 2-surface belong by the definition of number theoretic braids to these critical sub-manifolds. At the boundaries of CD associated with positive and negative energy parts of zero energy state in given time scale partonic two-surfaces belong to a fixed page of the Big Book whereas light-like 3-surface decomposes into regions corresponding to different values of Planck constant much like matter decomposes to several phases at thermodynamical criticality.

  4. The connection with Jones inclusions was originally a purely heuristic guess based on the observation that the finite groups characterizing Jones inclusion characterize also pages of the Big Book. The key observation is that Jones inclusions are characterized by a finite subgroup G SU(2) and that this group also characterizes the singular covering or factor spaces associated with CD or CP2 so that the pages of generalized imbedding space could indeed serve as correlates for Jones inclusions. The elements of the included algebra M are invariant under the action of G and M takes the role of complex numbers in the resulting non-commutative quantum theory.

  5. The understanding of quantum TGD at parton level led to the realization that the dynamics of Kähler action realizes finite measurement resolution in terms of finite number of modes of the induced spinor field. This automatically implies cutoffs to the representations of various super-conformal algebras typical for the representations of quantum groups closely associated with Jones inclusions. The Clifford algebra spanned by the fermionic oscillator operators would provide a realization for the factor space N/M of hyper-finite factors of type II1 identified as the infinite-dimensional Clifford algebra N of the configuration space and included algebra M determining the finite measurement resolution. The resulting quantum Clifford algebra has anti-commutation relations dictated by the fractionization of fermion number so that its unit becomes r=hbar/hbar0. SU(2) Lie algebra transforms to its quantum variant corresponding to the quantum phase q=exp(i2p/r).

  6. Jones inclusions appear as two variants corresponding to N:M < 4 and N:M=4. The tentative interpretation is in terms of singular G-factor spaces and G-coverings of M4 or CP2 in some sense. The alternative interpretation in terms of two geodesic spheres of CP2 would mean asymmetry between M4 and CP2 degrees of freedom.

  7. Number theoretic Universality suggests an answer why the hierarchy of Planck constants is necessary. One must be able to define the notion of angle -or at least the notion of phase and of trigonometric functions- also in p-adic context. All that one can achieve naturally is the notion of phase defined as root of unity and introduced by allowing algebraic extension of p-adic number field by introducing the phase if needed. In the framework of TGD inspired theory of consciousness this inspires a vision about cognitive evolution as the gradual emergence of increasingly complex algebraic extensions of p-adic numbers and involving also the emergence of improved angle resolution expressible in terms of phases exp(i2p/n) up to some maximum value of n. The coverings and factor spaces would realize these phases geometrically and quantum phases q naturally assignable to Jones inclusions would realize them algebraically. Besides p-adic coupling constant evolution based on hierarchy of p-adic length scales there would be coupling constant evolution with respect to hbar and associated with angular resolution.

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PART V: APPLICATIONS



Cosmology and Astrophysics in Many-Sheeted Space-Time

In this chapter the applications of TGD to astrophysics and cosmology are discussed. TGD inspired cosmology and astrophysics in their recent forms rely on an ontology differing dramatically from that of GRT based cosmologies. Zero energy ontology states that all physical states have vanishing net quantum numbers so that all matter is creatable from vacuum. The hierarchy of dark matter identified as macroscopic quantum phases labeled by arbitrarily large values of Planck constant is second aspect of the new ontology. The values of the gravitational Planck constant assignable to space-time sheets mediating gravitational interaction are gigantic. This implies that TGD inspired late cosmology might decompose into stationary phases corresponding to stationary quantum states in cosmological scales and critical cosmologies corresponding to quantum transitions changing the value of the gravitational Planck constant and inducing an accelerated cosmic expansion.

1. Does Equivalence Principle hold true in TGD Universe?

The understanding of Equivalence Principle in TGD framework has taken a long time.

  1. First came the conviction that coset representation for super-symplectic and super Kac-Moody algebras provides extremely general formulation of Equivalence Principle in which inertial and gravitational four-momenta are replaced with Super Virasoro generators of two algebras whose differences annihilate physical states. This idea came for years before becoming aware of its importance and I simply forgot it.

  2. Next came the realization of the fundamental role of number theoretical compactification providing a number theoretical interpretation of M4×CP2 and thus also of standard model quantum numbers. This lead to the identification of the preferred extremals of Kähler action and to the formulation of quantum TGD in terms of second quantized induced spinors fields. One of conclusion was that dimensional reduction for preferred extremals of Kähler action- if they have the properties required by theoretic compactification- leads to string model with string tension which is however not proportional to the inverse of Newton's constant but to Lp2, p-adic length scale squared and thus gigantic. The connection between gravitational constant and Lp2 comes from an old argument that I discovered about two decades ago and which allowed to predict the value of Kähler coupling strength by using as input electron mass and p-adic mass calculations. In this framework the role of Planck length as a fundamental length scale is taken by CP2 size so that Planck length scale loses its magic role as a length scale in which usual views about space-time geometry cease to hold true.

  3. The next step was the realization that zero energy ontology allows to avoid the paradox implied in positive energy ontology by the fact that gravitational energy is not conserved but inertial energy identified as Noether charge is. Energy conservation is always in some length scale in zero energy ontology.

  4. As a matter fact, there was still one step. I had to become fully aware that the identification of gravitational four-momentum in terms of Einstein tensor makes sense only in long length scales. This is of course trivial but for some reason I did not realize that this fact resolves the paradoxes associated with objects like cosmic strings.

The construction of quantum theory leads naturally to zero energy ontology stating that everything is creatable from vacuum. Zero energy states decompose into positive and negative energy parts having identification as initial and final states of particle reaction in time scales of perception longer than the geometro-temporal separation T of positive and negative energy parts of the state. If the time scale of perception is smaller than T, the usual positive energy ontology applies.

In zero energy ontology inertial four-momentum is a quantity depending on the temporal time scale T used and in time scales longer than T the contribution of zero energy states with parameter T1 < T to four-momentum vanishes. This scale dependence alone implies that it does not make sense to speak about conservation of inertial four-momentum in cosmological scales. Hence it would be in principle possible to identify inertial and gravitational four-momenta and achieve strong form of Equivalence Principle. It however seems that this is not the correct approach to follow.

Dark matter revolution with levels of the hierarchy labeled by values of Planck constant forces a further generalization of the notion of imbedding space and thus of space-time. One can say, that imbedding space is a book like structure obtained by gluing together infinite number of copies of the imbedding space like pages of a book: two copies characterized by singular discrete bundle structure are glued together along 4-dimensional set of common points. These points have physical interpretation in terms of quantum criticality. Particle states belonging to different sectors (pages of the book) can interact via field bodies representing space-time sheets which have parts belonging to two pages of this book.

4. Quantum criticality

TGD Universe is quantum counterpart of a statistical system at critical temperature. Quite recently it became clear that quantum criticality at the space-time level means the vanishing of second variation of Kähler action for preferred extremals- at least for deformations representing dynamical symmetries.

As a consequence, topological condensate is expected to possess hierarchical, fractal like structure containing topologically condensed 3-surfaces with all possible sizes. Both Kähler magnetized and Kähler electric 3-surfaces ought to be important and string like objects indeed provide a good example of Kähler magnetic structures important in TGD inspired cosmology. In particular space-time is expected to be many-sheeted even at cosmological scales and ordinary cosmology must be replaced with many-sheeted cosmology. The presence of vapor phase consisting of free cosmic strings containing topologically condensed fermions is second crucial aspect of TGD inspired cosmology.

Quantum criticality of TGD Universe, which corresponds to the vanishing of second variation of Kähler action for preferred extremals - at least of the variations related to dynamical symmetries- supports the view that many-sheeted cosmology is in some sense critical. Criticality in turn suggests fractality. Phase transitions, in particular the topological phase transitions giving rise to new space-time sheets, are (quantum) critical phenomena involving no scales. If the curvature of the 3-space does not vanish, it defines scale: hence the flatness of the cosmic time=constant section of the cosmology implied by the criticality is consistent with the scale invariance of the critical phenomena. This motivates the assumption that the new space-time sheets created in topological phase transitions are in good approximation modelable as critical Robertson-Walker cosmologies for some period of time at least.

These phase transitions are between stationary quantum states having stationary cosmologies as space-time correlates: also these cosmologies are determined uniquely apart from single parameter.

TGD allows global imbedding of subcritical cosmologies. A partial imbedding of one-parameter families of critical and overcritical cosmologies is possible. The infinite size of the horizon for the imbeddable critical cosmologies is in accordance with the presence of arbitrarily long range fluctuations at criticality and guarantees the average isotropy of the cosmology. Imbedding is possible for some critical duration of time. The parameter labeling these cosmologies is scale factor characterizing the duration of the critical period. These cosmologies have the same optical properties as inflationary cosmologies. Critical cosmology can be regarded as a 'Silent Whisper amplified to Bang' rather than 'Big Bang' and transformed to hyperbolic cosmology before its imbedding fails. Split strings decay to elementary particles in this transition and give rise to seeds of galaxies. In some later stage the hyperbolic cosmology can decompose to disjoint 3-surfaces. Thus each sub-cosmology is analogous to biological growth process leading eventually to death.

The critical cosmologies can be used as a building blocks of a fractal cosmology containing cosmologies containing ... cosmologies. p-Adic length scale hypothesis allows a quantitative formulation of the fractality. Fractal cosmology predicts cosmos to have essentially same optic properties as inflationary scenario but avoids the prediction of unknown vacuum energy density. Fractal cosmology explains the paradoxical result that the observed density of the matter is much lower than the critical density associated with the largest space-time sheet of the fractal cosmology. Also the observation that some astrophysical objects seem to be older than the Universe, finds a nice explanation.

Cosmic strings are the basic building blocks of TGD inspired cosmology and all structures including large voids, galaxies, stars, and even planets can be seen as pearls in a cosmic fractal necklaces consisting of cosmic strings containing smaller cosmic strings linked around them containing... During cosmological evolution the cosmic strings are transformed to magnetic flux tubes with smaller Kähler string tension and these structures are also key players in TGD inspired quantum biology.

The observed large voids would contain galactic cosmic strings at their boundaries. These voids would participate cosmic expansion only in average sense. During stationary periods the quantum states would be modelable using stationary cosmologies and during phase transitions increasing gravitational Planck constant and thus size of the large void they critical cosmologies would be the appropriate description. The acceleration of cosmic expansion predicted by critical cosmologies can be naturally assigned with these periods.

7. Quantum astrophysics

The identification of dark matter as a hierarchy of macroscopic quantum phases labeled by the value of Planck constant transforms astrophysics to quantum astrophysics. Only two examples about numerous applications are discussed in this chapter.

  1. Dark black holes with gigantic value of Planck constant in anyonic state in which charges of particles are fractional and confined to an anyonic surface representing the analog of black hole horizon is a natural prediction of TGD. The standard view about black holes as highly entropic objects must be given up since black hole entropy is proportional to 1/hbar and would be of order unity for these objects. As a matter fact, all dark matter could be confined inside this kind of surfaces accompanying also planetary orbits.

  2. A further fascinating possibility -which could be inspired by the anyonic vision - is that the evidence for Bohr orbit quantization of planetary orbits could have interpretation in terms of gigantic Planck constant for underlying dark matter so that macroscopic and -temporal quantum coherence would be possible in astrophysical length scales manifesting itself in many manners: say as preferred directions of quantization axis (perhaps related to the CMB anomaly) or as anomalously low dissipation rates. Since the gravitational Planck constant hbargr=GM1m/v0, v0=2-11 for the inner planets, is proportional to the product of the gravitational masses of interacting systems, it must be assigned to the field body of the two systems and characterizes the interaction between systems rather than systems themselves. This observation applies quite generally and each field body of the system (em, weak, color, gravitational) is characterized by its own Planck constant.

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Elementary Particle Vacuum Functionals

Genus-generation correspondence is one of the basic ideas of TGD approach. In order to answer various questions concerning the plausibility of the idea, one should know something about the dependence of the elementary particle vacuum functionals on the vibrational degrees of freedom for the boundary component. The construction of the elementary particle vacuum functionals based on Diff invariance, 2-dimensional conformal symmetry, modular invariance plus natural stability requirements indeed leads to an essentially unique form of the vacuum functionals and one can understand why g >2 bosonic families are experimentally absent and why lepton numbers are conserved separately.

An argument suggesting that the number of the light fermion families is three, is developed. The argument goes as follows. Elementary particle vacuum functionals represent bound states of g handles and vanish identically for hyper-elliptic surfaces having g > 2. Since all g≤ 2 surfaces are hyper-elliptic, g≤ 2 and g > 2 elementary particles cannot appear in same non-vanishing vertex and therefore decouple. The g>2 vacuum functionals not vanishing for hyper-elliptic surfaces represent many particle states of g≤ 2 elementary particle states being thus unstable against the decay to g≤ 2 states. The failure of Z2 conformal symmetry for g>2 elementary particle vacuum functionals could in turn explain why they are heavy: this however not absolutely necessary since these particles would behave like dark matter in any case.

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Massless States and Particle Massivation

In this chapter the goal is to summarize the recent theoretical understanding of the spectrum of massless particles and particle massivation in TGD framework. After a summary of the recent phenomenological picture behind particle massivation the notions of number theoretical compactification and number theoretical braid are introduced and the construction of quantum TGD at parton level in terms of second quantization of modified Dirac action is described. The recent understanding of super-conformal symmetries are analyzed in detail. TGD color differs in several respect from QCD color and a detailed analysis of color partial waves associated with quark and lepton chiralities of imbedding space spinors fields is carried out with a special emphasis given to the contribution of color partial wave to mass squared of the fermion. The last sections are devoted to p-adic thermodynamics and to a model providing a formula for the modular contribution to mass squared.

Although the basic predictions of p-adic mass calculations were known almost 15 years ago, the justification of the basic assumptions from basic principles of TGD (and also the discovery of these principles!) has taken a considerable time. Particle massivation can be regarded as a generation of thermal conformal weight identified as mass squared and due to a thermal mixing of a state with vanishing conformal weight with those having higher conformal weights. The observed mass squared is not p-adic thermal expectation of mass squared but that of conformal weight so that there are no problems with Lorentz invariance.

One can imagine several microscopic mechanisms of massivation. The following proposal is the winner in the fight for survival between several competing scenarios.

  1. The original observation was that the pieces of CP2 type vacuum extremals representing elementary particles have random light-like curve as an M4 projection so that the average motion correspond to that of massive particle. Light-like randomness gives rise to classical Virasoro conditions. This picture generalizes since the basic dynamical objects are light-like but otherwise random 3-surfaces. Fermions are identified as light-like 3-surfaces at which the signature of induced metric of deformed CP2 type extremals changes from Euclidian to the Minkowskian signature of the background space-time sheet. Gauge bosons and Higgs correspond to wormhole contacts with light-like throats carrying fermion and antifermion quantum numbers. Gravitons correspond to pairs of wormhole contacts bound to string like object by the fluxes connecting the wormhole contacts. The randomness of the light-like 3-surfaces and associated super-conformal symmetries justify the use of thermodynamics and the question remains why this thermodynamics can be taken to be p-adic. The proposed identification of bosons means enormous simplification in thermodynamical description since all calculations reduced to the calculations to fermion level.

  2. The fundamental parton level description of TGD is based on almost topological QFT for light-like 3-surfaces. Dynamics is constrained by the requirement that CP2 projection is for extremals of Chern-Simons action 2-dimensional and for off-shell states light-likeness is the only constraint. As a matter fact, the basic theory relies on the modified Dirac action associated with Chern-Simons action and Kähler action in the sense that the generalizes eigenmodes of C-S Dirac operator correspond to the zero modes of Kähler action localized to the light-like 3-surfaces representing partons. In this manner the data about the dynamics of Kähler action is feeded to the eigenvalue spectrum. Eigenvalues are interpreted as square roots of ground state conformal weights.

  3. The symmetries respecting light-likeness property give rise to Kac-Moody type algebra and super-symplectic symmetries emerge also naturally as well as N=4 character of super-conformal invariance. The coset construction for super-symplectic Virasoro algebra and Super Kac-Moody algebra identified in physical sense as sub-algebra of former implies that the four-momenta assignable to the two algebras are identical. The interpretation is in terms of the identity of gravitational inertial masses and generalization of Equivalence Principle.

  4. Instead of energy, the Super Kac-Moody Virasoro (or equivalently super-symplectic) generator L0 (essentially mass squared) is thermalized in p-adic thermodynamics (and also in its real version assuming it exists). The fact that mass squared is thermal expectation of conformal weight guarantees Lorentz invariance. That mass squared, rather than energy, is a fundamental quantity at CP2 length scale is also suggested by a simple dimensional argument (Planck mass squared is proportional to hbar so that it should correspond to a generator of some Lie-algebra (Virasoro generator L0!)).

  5. By Equivalence Principle the thermal average of mass squared can be calculated either in terms of thermodynamics for either super-symplectic of Super Kac-Moody Virasoro algebra and p-adic thermodynamics is consistent with conformal invariance.

  6. A long standing problem has been whether coupling to Higgs boson is needed to explain gauge boson masses. It has turned out that p-adic thermodynamics is enough. From the beginning it was clear that is that ground state conformal weight is negative. Only quite recently it became clear that the ground state conformal weight need not be a negative integer. The deviation Δh of the total ground state conformal weight from negative integer gives rise to Higgs type contribution to the thermal mass squared and dominates in case of gauge bosons for which p-adic temperature is small. In the case of fermions this contribution to the mass squared is small. Higgs vacuum expectation is naturally proportional to Δh so that the coupling to Higgs apparently causes gauge boson massivation. The interpretation is that the effective metric defined by the modified gamma matrices associated with Kähler action has Euclidian signature. This implies that the eigenvalues of the modified Dirac operator are purely imaginary and analogous to cyclotron energies so that in the first approximation smallest conformal weights are of form h = -n-1/2 and for n=0 one obtains the ground state conformal weight h=-1/2 conjectured earlier. One cannot exclude the possibility of complex eigenvalues of DC-S.

  7. There is also modular contribution to the mass squared which can be estimated using elementary particle vacuum functionals in the conformal modular degrees of freedom of the partonic 2-surface. It dominates for higher genus partonic 2-surfaces. For bosons both Virasoro and modular contributions seem to be negligible and could be due to the smallness of the p-adic temperature.

An important question concerns the justification of p-adic thermodynamics.

  1. The underlying philosophy is that real number based TGD can be algebraically continued to various p-adic number fields. This gives justification for the use of p-adic thermodynamics although the mapping of p-adic thermal expectations to real counterparts is not completely unique. The physical justification for p-adic thermodynamics is effective p-adic topology characterizing the 3-surface: this is the case if real variant of light-like 3-surface has large number of common algebraic points with its p-adic counterpart obeying same algebraic equations but in different number field.

  2. The most natural option is that the descriptions in terms of both real and p-adic thermodynamics make sense and are consistent. This option indeed makes since the number of generalized eigen modes of modified Dirac operator is finite. The finite number of fermionic oscillator operators implies an effective cutoff in the number conformal weights so that conformal algebras reduce to finite-dimensional algebras. The first guess would be that integer label for oscillator operators becomes a number in finite field for some prime. This means that one can calculate mass squared also by using real thermodynamics but the consistency with p-adic thermodynamics gives extremely strong number theoretical constraints on mass scale. This consistency condition allows also to solve the problem how to map a negative ground state conformal weight to its p-adic counterpart. Negative conformal weight is divided into a negative half odd integer part plus positive part Δh, and negative part corresponds as such to p-adic integer whereas positive part is mapped to p-adic number by canonical identification.

p-Adic thermodynamics is what gives to this approach its predictive power.

  1. p-Adic temperature is quantized by purely number theoretical constraints (Boltzmann weight exp(-E/kT) is replaced with pL0/Tp, 1/Tp integer) and fermions correspond to Tp=1 whereas Tp=1/n, n > 1, seems to be the only reasonable choice for gauge bosons.

  2. p-Adic thermodynamics forces to conclude that CP2 radius is essentially the p-adic length scale R ~ L and thus of order R @ 103.5 {hbar G} and therefore roughly 103.5 times larger than the naive guess. Hence p-adic thermodynamics describes the mixing of states with vanishing conformal weights with their Super Kac-Moody Virasoro excitations having masses of order 10-3.5 Planck mass.

The predictions of the general theory are consistent with the earliest mass calculations, and the earlier ad hoc parameters disappear. In particular, optimal lowest order predictions for the charged lepton masses are obtained and photon, gluon and graviton appear as essentially massless particles.

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